For an open complete Riemannian manifold $M$ with non-negative sectional curvature, the Busemann function defined below is a convex exhaustion function (by Cheeger-Gromoll's proof of soul theorem)

The Buesemann function: $$b(x)=\sup_{\gamma} b_{\gamma}(x)$$ where the $sup$ is taken over all rays from a given point and $b_{\gamma}$ is the Busemann function associated with ray $\gamma$:

$$b_{\gamma}(x)=\lim_{t\to \infty}(t-d(x, \gamma(t))$$

A function $f:M\to \mathbb [a,\infty)$ is called an exhaustion function on $M$ if its sublevel set $\Omega_c:=f^{-1}((-\infty, c])$ is compact for all $c$ and $M=\cup_c \Omega_c$.

I am wondering that when we assume Ricci curvature is non-negative, is there any example where the Busemann function $b$ is not an exhaustion? (In this case the Busemann function is subharmonic.)